K-energy on polarized compactifications of Lie groups. (English) Zbl 1392.53076
Summary: In this paper, we study Mabuchi’s K-energy on a compactification \(M\) of a reductive Lie group \(G\), which is a complexification of its maximal compact subgroup \(K\). We give a criterion for the properness of K-energy on the space of \(K \times K\)-invariant Kähler potentials. In particular, it turns to give an alternative proof of Delcroix’s theorem for the existence of Kähler-Einstein metrics in case of Fano manifolds \(M\). We also study the existence of minimizers of K-energy for general Kähler classes of \(M\).
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 58J05 | Elliptic equations on manifolds, general theory |
| 19L10 | Riemann-Roch theorems, Chern characters |
| 53C30 | Differential geometry of homogeneous manifolds |
References:
| [1] | Alexeev, V.; Brion, M., Stable reductive varieties I: affine varieties, Invent. Math., 157, 227-274, (2004) · Zbl 1068.14003 |
| [2] | Alexeev, V.; Brion, M., Stable reductive varieties II: projective case, Adv. Math., 184, 382-408, (2004) · Zbl 1080.14017 |
| [3] | Alexeev, V.; Katzarkov, L., On K-stability of reductive varieties, Geom. Funct. Anal., 15, 297-310, (2005) · Zbl 1122.32016 |
| [4] | Azad, H.; Loeb, J., Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. (N.S.), 3, 4, 365-375, (1992) · Zbl 0777.32008 |
| [5] | Berman, R.; Berndtsson, B., Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics · Zbl 1376.32028 |
| [6] | Cao, H.; Tian, G.; Zhu, X., Kähler-Ricci solitons on compact complex manifolds with \(C_1(M) > 0\), Geom. Funct. Anal., 15, 697-719, (2005) · Zbl 1084.53035 |
| [7] | Chen, X.; Cheng, J., On the constant scalar Kähler metrics, general automorphism group |
| [8] | Chen, X.; Donaldson, S.; Sun, S., Kähler-Einstein metrics on Fano manifolds I-III, J. Amer. Math. Soc., 28, 183-278, (2015) · Zbl 1312.53096 |
| [9] | Chen, X.; Tian, G., Ricci flow on Kähler-Einstein manifolds, Duke Math. J., 131, 17-73, (2006) · Zbl 1097.53045 |
| [10] | Darvas, T.; Rubinstein, Y., Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc., 30, 347-387, (2017) · Zbl 1386.32021 |
| [11] | Delcroix, T., Log canonical thresholds on group compactifications · Zbl 1393.14050 |
| [12] | Delcroix, T., K-stability of Fano spherical varieties · Zbl 1473.14098 |
| [13] | Delcroix, T., Kähler-Einstein metrics on group compactifications, Geom. Funct. Anal., 27, 78-129, (2017) · Zbl 1364.32017 |
| [14] | Donaldson, S., Scalar curvature and stability of toric varieties, J. Differential Geom., 62, 289-348, (2002) · Zbl 1074.53059 |
| [15] | Futaki, A., On a character of the automorphism group of a compact complex manifold, Invent. Math., 87, 655-660, (1987) · Zbl 0612.53044 |
| [16] | Guillemin, V., Kähler structures on toric varieties, J. Differential Geom., 40, 285-309, (1994) · Zbl 0813.53042 |
| [17] | Helgason, S., Differential geometry, Lie groups, and symmetric spaces, (1978), Academic Press, Inc. New York-London · Zbl 0451.53038 |
| [18] | Hu, Z.; Yan, K., The Weyl integration model for KAK decomposition of reductive Lie groups |
| [19] | Knapp, A., Representation theory of semisimple groups, (1986), Princeton Univ. Press Princeton, NJ · Zbl 0604.22001 |
| [20] | Knapp, A., Lie groups beyond an introduction, (2002), Birkhäuser Boston, Inc. Boston · Zbl 1075.22501 |
| [21] | Tian, G., On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., 101, 101-172, (1990) · Zbl 0716.32019 |
| [22] | Tian, G., Kähler-Einstein metrics with positive scalar curvature, Invent. Math., 130, 1-37, (1997) · Zbl 0892.53027 |
| [23] | Tian, G., Existence of Einstein metrics on Fano manifolds, (Jeff Cheeger Anniversary Volume: Metric and Differential Geometry, Progress in Math., vol. 297, (2012)), 119-162 · Zbl 1250.53044 |
| [24] | Tian, G., K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math., 68, 1085-1156, (2015) · Zbl 1318.14038 |
| [25] | Tian, G.; Zhu, X., A new holomorphic invariant and uniqueness of Kähler-Ricci solitons, Comment. Math. Helv., 77, 297-325, (2002) · Zbl 1036.53053 |
| [26] | Tian, G.; Zhu, X., Convergence of Kähler-Ricci flow, J. Amer. Math. Soc., 20, 675-699, (2007) · Zbl 1185.53078 |
| [27] | Trudinger, N.; Wang, X., The affine plateau problem, J. Amer. Math. Soc., 18, 253-289, (2005) · Zbl 1229.53049 |
| [28] | Wang, F.; Zhou, B.; Zhu, X., Modified Futaki invariant and equivariant Riemann-Roch formula, Adv. Math., 289, 1205-1235, (2016) · Zbl 1342.14106 |
| [29] | Wang, X.; Zhu, X., Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math., 188, 87-103, (2004) · Zbl 1086.53067 |
| [30] | Zhou, B., The first boundary value problem for Abreu’s equation, Int. Math. Res. Not., 7, 1439-1484, (2012) · Zbl 1242.35101 |
| [31] | Zhou, B., Variational solutions to extremal metrics on toric surfaces, Math. Z., 283, 1011-1031, (2016) · Zbl 1351.53087 |
| [32] | Zhou, B.; Zhu, X., Relative K-stability and modified K-energy on toric manifolds, Adv. Math., 219, 1327-1362, (2008) · Zbl 1153.53030 |
| [33] | Zhou, B.; Zhu, X., Minimizing weak solutions for Calabi’s extremal metrics on toric manifolds, Calc. Var., 32, 191-217, (2008) · Zbl 1141.53061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
